Domain wall effects in ferromagnet-superconductor structures. Igor S. Burmistrov1, 2 and Nikolai M. Chtchelkatchev1, 3, 4, ∗
arXiv:cond-mat/0501210v1 [cond-mat.mes-hall] 10 Jan 2005
1
L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, 117940 Moscow, Russia 2 Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018XE Amsterdam, The Netherlands 3 Institute for High Pressure Physics, Russian Academy of Sciences, Troitsk 142092, Moscow Region, Russia 4 Moscow Institute of Physics and Technology, Moscow 141700, Russia We investigate how domain structure of the ferromagnet in superconductor-ferromagnet heterostructures may change their transport properties. We calculate the distribution of current in the superconductor induced by magnetic field of Bloch domain walls, find the “lower critical” magnetization of the ferromagnet that provides vortices in the superconductor. PACS numbers: 05.60.Gg, 74.50.+r, 74.81.-g, 75.70.-i
Superconductivity and ferromagnetism are two competing phenomena: while the first prefers antiparallel spin orientation of electrons in Cooper pairs, the second forces the spins to be aligned in parallel. Their coexistence in one and same material or their interaction in spatially separated materials leads to a number of new interesting phenomena, for example, π-state of superconductor (S) – ferromagnet (F) – superconductor (SFS) Josephson junctions,1,2,3,4,5,6,7,8 highly nonmonotonic dependence of the critical temperature Tc of a SF bilayer as a function of the ferromagnet thickness9 and so on. Recent investigations of SF bilayers showed that their transport properties often strongly depend on the interplay between magnetic structure of the ferromagnet and superconductivity.10,11,12,13,14,15,16,17,18 In particular, it was argued that due to ferromagnetic domains vortices may appear in the superconducting film of the SF bilayer and the domain configuration, in turn, may depend on the vortices.12 Recently, generation of vortices by magnetic texture of the ferromagnet in SF junctions was demonstrated experimentally.18 In a number of experiments dealing with Tc of SF bilayers, or Josephson effect in SFS structures1 the domain magnetizations were parallel to the SF interface. Ferromagnets used in the experiments were often dilute with the exchange field comparable to the superconducting gap and small domain size [smaller or comparable to the bulk superconductor screening length] and broad domain walls.1 In this paper we try to put a step forward the answer to the question, how domain structure of the ferromagnet in SF heterostructures may change their transport properties. In the first part of the paper we discuss the junctions where S and F are weakly coupled (there is insulator layer in-between such that there is no proximity effect) and magnetizations of the domains are parallel to the SF interface. We find the distribution of current in the superconductor induced by magnetic field of the domain walls and the “lower critical” magnetization of the ferromagnet that provides vortices in the superconductor. In the end of the paper we estimate the critical temperature in strongly coupled SF bilayer when the proximity effect is strong. In this paper we do not consider the rearrangement of the domain configuration due
z x y
δ
F S
FIG. 1: The SF junction. A sketch of a Bloch domain wall. The magnetization rotates according to Eq. (2). Magnetization in the center of the domain wall is perpendicular to the S film.
to the superconductor12 though we mention the superconductor induced transitions between Bloch and Neel domain wall types. The point is that the crystal structure of the ferromagnets in experiments1 was not perfect. Experimental data suggests that defects, dislocations in the lattice that appear during lithography process stick domain configuration. The domain texture of the F film is described by the following magnetization (see Fig. 1) M = M θ (z) θ (dF − z)
∞ X
(−1)j m(x − jL).
(1)
j=−∞
where vector m(x) rotates as follows19 mx = 0,
my = tanh(x/δ),
mz = −1/ cosh(x/δ). (2)
The vector potential A satisfies the Maxwell-Londons equation ∇ × (∇ × A) + Aλ−2 L θ (−z) θ (z + dS ) = 4π∇ × M (3) that should be supplemented by the standard boundary conditions of continuity of A and ∂A/∂z.19 By solving Eq. (3) with a help of the Fourier transformation we can find the distribution of the magnetic field
2 B = ∇ × A in the entire space.20 In agreement with general expectations the z component of the magnetization in the F film that collects at domain walls results in the current flow in the S film. It is convenient to de-
Jy (x) = −2πcM
∞ X
Qn sinh Qn dS + qn (cosh Qn dS − 1) sin qn x [1 − exp(−qn dF )] , λ2L L n=0 Qn cosh( π2 qn δ) (Q2n + qn2 ) sinh Qn dS + 2Qn qn cosh Qn dS δ
where c stands q for the speed of light, qn = π(2n + 1)/L and Qn = qn2 + λ−2 L . Eq. (4) constitutes one of the principal results of the present paper. It allows to compute the distribution of the current flow in the S film for a general set of parameters dS , dF , δ, L and λL . Below we shall analyze two the most interesting cases of thick (dS , dF ≫ λL ) and thin (dS , dF ≪ λL ) SF bilayer. Thick SF bilayer. Eq. (4) can be drastically simplified provided dS , dF ≫ λL such that the current Jy (x) becomes independent on the widths dS and dF of the S and F films. It is given as Jy (x) = −cM
∞ 2πδ X sin qn x 1 . π 2 λL L n=0 cosh 2 qn δ Qn (Qn + qn )
(5)
In order to understand the distribution of the current Jy (x) as determined by Eq. (5) we shall first analyze the case of a single domain wall. Taking the limit L → ∞ in Eq. (5), we obtain the following result for the current in the presence of a single domain wall in the F film Z ∞ sin λxL ω Jy (x) δ 1 dω √ √ . (6) =− πωδ 2 cM λL 0 1 + ω cosh 2λL ω + 1 + ω 2 The distribution of the current Jy (x) is governed by the single parameter πδ/(2λL ) as it is shown in Fig. 2. If the width δ of the domain wall is small compared to the Londons penetration length λL , πδ/(2λL ) ≪ 1, we find the distribution of the current as h i δ |x| K |x| − 1 , |x| ≫ δ, 1 Jy (x) = cM xxδ λL πδ λL (7) 2 ln , |x| ≪ δ, 2λL
fine the current averaged R 0 over the thickness of the S film, Jy (x) = −c/(4πλ2L ) −dS Ay dz. Then we obtain
2λL
where K1 (x) is the modified Bessel function of the second kind. In the opposite case of the thick domain wall, πδ/(2λL ) ≫ 1, we obtain λL tanh xδ x 2 x 1 Jy (x) + = tanh − Im ψ +i , (8) cM δ π 4 2πδ δ cosh xδ
where ψ(x) denotes the digamma function. According to Eqs. (7) and (8) the current Jy (x) behaves linearly with x for x ≪ δ and decays as a power law for large x. The current distribution Jy (x) is spread
(4)
on the length Ls ∝ max{δ, λL } from the origin whereas its maximal value Jym ∝ cM δ/Ls = cM min{1, δ/λL }. Now we turn back to the general case of multi domain wall structure in the F film that corresponds to a finite size L of domains. We have evaluated the sum in Eq. (5) numerically and present results for the current distribution in Fig. 3. While λL remains small compared with L the profile of Jy (x) corresponds to almost independent current distributions near each domain wall that results in distinctive two maximum structure as shown in Fig. 3. When λL becomes of the order of L the two maximum structure transforms into sinusoidal profile with the maximum exactly in the middle of a domain. Thin SF bilayer. In the case of the thin SF bilayer, dS , dF ≪ λL , by expanding the general expression (4) in powers of dS and dF , we find ∞ 2πδ X sin qn x qn Jy (x) = −cM dF . L n=0 cosh π2 qn δ 1 + 2qn λ
(9)
Here λ = λ2L /dS usually referred as the effective penetration length.21 We shall first analyze the case of a single domain wall again. In the limit L → ∞ we obtain from Eq. (9) Z x ω dF δ ∞ ω sin 2λ Jy (x) = −cM 2 . (10) dω πδ 4λ 0 1 + ω cosh 4λ ω The distribution of the current Jy (x) is presented in Fig. 4 for different values of the parameter πδ/(4λ). If the domain wall is thin, πδ/(4λ) ≪ 1, Eq. (10) yields x dF h 2 Jy (x) tanh − Im ψ = cM 2λ δ π
x 1 +i 4 2πδ
+
i δ g(x) , 2λ (11)
where g(x) =
(
π x 2 sign x cos 2λ x δ,
x + f ( 2λ ), |x| ≫ δ, |x| ≪ δ.
(12)
Here f (x) = sin x ci(x) − cos x si(x) with ci(x) and si(x) being the cosine and sine integral functions. In the opposite case πδ/(4λ) ≫ 1 the current distribution Jy (x) is given as dF tanh x/δ Jy (x) =− . cM δ cosh x/δ
(13)
3 Π∆ 2Λ 0.1 L 0.5 1 2 4
0.2 0 -0.2 -0.4
0.2 0 -0.2 -0.4
-15 -10
-5
0 xΛL
5
10
15
20
FIG. 2: The case d ≫ λL . The plot of Jy (x)/(cM ) as a function of x/λL for πδ/(2λL ) = 0.1, 0.5, 1, 2, 4.
-10
0 -0.2
0 x2Λ
5
10
ΛL 0.05 0.1 0.25 0.5 0.75 1
0.4
Jy HxLcM
0.2
-5
FIG. 4: The case d ≪ λL . The plot of 2λJy (x)/(cM dF ) as a function of x/(2λ) for πδ/(4λ) = 0.2, 0.5, 1, 2, 4.
ΛL L 0.005 0.01 0.05 0.1 0.25 0.5
0.4
Jy HxLcM
Π∆ 4Λ 0.2 0.5 1 2 4
0.4 2ΛJy HxLcMdF
Jy HxLcM
0.4
0.2 0 -0.2
-0.4
-0.4 -0.4
-0.2
0 xL
0.2
0.4
FIG. 3: The case d ≫ λL . The plot of Jy (x)/(cM ) as a function of x/L for δ/L = 0.02 and λL /L = 0.005, 0.01, 0.05, 0.1, 0.25, 0.5.
Eqs. (11) and (13) proves that the Jy (x) increases linearly with x for x ≪ δ and decreases algebraically for large x. The position of the maximum of Jy (x) is situated at Ls ∝ δ and the value at the maximum Jym ∝ (dF /δ) min{1, δ/λ}. As one can see therefore the current distribution for the thin SF bilayer is qualitatively different from one for the thick SF bilayer. In the case of multi domain wall structure in the F layer we have performed evaluation of the sum in Eq. (9) numerically and have obtained the results for the current distribution Jy (x) presented in Fig. 5. We mention that the two maximum structure survives even for λ of the order of L for the thin SF bilayer. As known the lower critical field for a thin S film is much smaller than for the bulk superconductor. Therefore it is possible that even small magnetization collects at domain walls can induce a vortex in the thin S film.12 Let us assume that there is a single vortex in the the thin S film situated at x = 0. The magnetic field becomes a sum of the magnetic field induced by the domain walls and the magnetic field of the vortex. The free energy can be written as 2 Z B λ2 3 2 F = d r + |∇ × B| − MB , (14) 8π 8π
-0.4
-0.2
0 xL
0.2
0.4
FIG. 5: The case d ≪ λL . The plot of Jy (x)/(cM ) as a function of x/L for δ/L = 0.02, dF /L = 0.1 and λ/L = 0.05, 0.1, 0.25, 0.5, 0.75, 1.
where B denotes the total magnetic field. The difference F of the free energy for the state with the vortex and the free energy for the state without vortex is given as follows20 φ0 Hc 1 λ M F= , Mc = G(δ, λ, L), (15) Hc 1 λ 1 − 4π Mc 4π dF where φ0 = ch/(2e), Hc1 = (φ0 /4πλ2 ) ln λ/ξ is the lower critical field in the thin S film without the F film and " #−1 ∞ 2π X 1 δ G(δ, λ, L) = . (16) L n=0 cosh πq2n δ (1 + 2qn λ)2 The F becomes negative if M > Mc and vortices can proliferate in the S film until vortex-vortex interaction stops it or (that is more probable) the domain wall changes to Neel domain wall type to reduce the free energy. In the most interesting case of a single domain wall we find ( Hc1 λ 2λ/δ, πδ/4λ ≪ 1, (17) Mc = 4π dF 1 − 32Gλ/(π 2 δ), πδ/4λ ≫ 1, where G ≈ 0.916 stands for the Catalan constant.
4 (Note, that Mc given by Eq.(17) differs from estimates of Ref.12). In conclusion, domain wall effects in ferromagetsuperconductor structures are investigated. We find the distribution of current in the superconductor induced by magnetic field of Bloch domain walls, calculate the “lower critical” magnetization of the ferromagnet that provides vortices in the superconductor. We neglected above the proximity effect in SF structure assuming that the ferromagnet and the superconductor are weakly coupled. Below we discuss the case when S and F are strongly coupled. Consider a SF bilayer with a perfect SF boundary. When the superconductor and the ferromagnet are thin enough then the bilayer can be described as a “ferromagnetic superconductor” with effective parameters:22 the superconducting gap ∆eff , the (eff) exchange field Eex ... The superconductivity survives
∗ 1
2
3
4
5
6
7
8 9
10
Electronic address:
[email protected] For review see V. V. Ryazanov, V. A. Oboznov, A.S. Prokofiev et al, J. Of Low Temp. Phys. 136, 385 (2004). L. N. Bulaevskii, V. V. Kuzii, and A. A. Sobyanin, JETP Lett. 25, 290 (1977); A. V. Andreev, A. I. Buzdin, and R. M. Osgood, Phys. Rev. B 43, 10124 (1991); A. I. Buzdin, B. Vujicic, and M. Yu. Kupriyanov, Zh. Eksp. Teor. Fiz. 101, 231 (1992) [Sov. Phys. JETP 74, 124 (1992)]. A. V. Veretennikov, V. V. Ryazanov, V. A. Oboznov et al., Physica B 284-288, 495 (2000); V. V. Ryazanov, V. A. Oboznov, A. Yu. Rusanov et al., Phys. Rev. Lett. 86, 2427 (2001). T. Kontos, M. Aprili, J. Lesueur, et al, Phys. Rev. Lett. 89, 137007 (2002). N.M. Chtchelkatchev, W. Belzig, Yu.V. Nazarov, and C. Bruder, JETP Lett. 74, 323 (2001); N.M. Chtchelkatchev, Pis’ma v ZhETF vol.80, iss.12, pp.875-879(2004). Yu. S. Barash and I. V. Bobkova, Phys. Rev. B 65, 144502 (2002). M. Bozovic, and Z. Radovic, Phys. Rev. B 66, 134524 (2002); Z. Radovic, N. Lazarides, and N. Flytzanis, Phys. Rev. B 68, 014501 (2003). C.W.J. Beenakker, Phys. Rev. Lett. 67, 3836 (1991). Ya.V. Fominov, N.M. Chtchelkatchev, and A.A. Golubov, JETP Lett. 74, 96 (2001). L.N. Bulaevskii and E.M. Chudnovsky, Phys. Rev. B 63,
(eff)
(0)
(0)
in this system until Eex < ∆eff , where ∆eff is the gap (eff) at Eex = 0.22 Domain wall structure of the ferromagnet makes the effective exchange field nonhomogeneous. (eff) We find that if Eex changes its sign on scales of the order of ξ0 or smaller q then superconductivity in the bi(eff)
(0)
(eff)
layer survives at h(Eex )2 i > ∆eff , where h(Eex )2 i is the average square of the effective exchange field over the sample.16
We are grateful to V. Ryazanov for stimulating discussions and also thank RFBR Project No. 03-02-16677, 04-02-08159 and 02-02-16622, the Russian Ministry of Science, the Netherlands Organization for Scientific Research NWO, CRDF, Russian Science Support foundation and State Scientist Support foundation (Project No. 4611.2004.2).
11 12
13
14
15
16
17 18 19
20 21 22
012502 (2000). E.B. Sonin, Phys. Rev. B 66, R100504 (2002). S. Erdin, A.F. Kayali, I. F. Lyuksyutov, and V. L.Pokrovsky, Phys. Rev. B 66, 014414 (2002); V.L. Pokrovsky and H. Wei, Phys.Rev. B 69, 104530 (2004); Luksyitov and V.L Pokrovsky, Mod. Phys. Lett. 14, 409 (2000). G. Deutscher and D. Feinberg, Appl. Phys. Lett. 76, 487; R. Melin, J. Phys.: Condens. Matter 13, 6445 (2001). R.J. Kinsey, G. Burnell, and M.J. Blamir, IEEE Trans. Appl. Superc. 11, 904 (2001). A. Kadigrobov, R.I. Shekhter, and M. Jonson, Europhys. Lett. 54, 394 (2001); F.S. Bergeret, A.F. Volkov, and K.B. Efetov, Phys. Rev. Lett. 86, 4096 (2001). N.M. Chtchelkatchev and I.S. Burmistrov, Phys. Rev. B 68, R140501 (2003). R. Melin, and S. Peysson, Phys. Rev. B 68, 174515 (2003). V. V. Ryazanov et al., JETP Lett. 77, 39 (2003). L.D. Landau and E.M. Lifshitz, in Course in Theoretical Physics, Vol. 8 (Pergamon Press, Oxford, 1984). I.S. Burmistrov and N.M. Chtchelkatchev, in preparation. J. Pearl, J. Appl. Phys. Lett. 5, 65 (1964). F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Phys. Rev. Lett. 86, 3140 (2001); Phys. Rev. B 64, 134506 (2001).
